5 In this question you must show detailed reasoning.
Fig. 12 shows part of the graph of .
The tangent to the curve at the point meets the -axis at A and meets the -axis
at B .
Ois the origin.
Find the exact area of the triangle OAB .

Ask by Cervantes Whittaker. in the United Kingdom

Nov 01,2024

Question

5 In this question you must show detailed reasoning.
Fig. 12 shows part of the graph of .
The tangent to the curve at the point meets the -axis at A and meets the -axis
at B .
Ois the origin.
Find the exact area of the triangle OAB .

Ask by Cervantes Whittaker. in the United Kingdom

Nov 01,2024

UpStudy AI · Accepted Answer

To find the area of triangle OAB formed by the origin O, point A where the tangent meets the x-axis, and point B where the tangent meets the y-axis, we will follow these steps:

1. **Find the derivative of the function** to determine the slope of the tangent line at the point $$ \left(2, \frac{17}{4}\right) $$.
2. **Use the point-slope form of the equation of the tangent line** to find the equations of the lines that meet the axes.
3. **Determine the coordinates of points A and B**.
4. **Calculate the area of triangle OAB** using the coordinates of points O, A, and B.

### Step 1: Find the derivative

The function is given by:

$$
y = x^2 + \frac{1}{x^2}
$$

To find the derivative $$ y' $$:

$$
y' = \frac{d}{dx}(x^2) + \frac{d}{dx}\left(\frac{1}{x^2}\right) = 2x - \frac{2}{x^3}
$$

Now, we evaluate the derivative at $$ x = 2 $$:

$$
y'(2) = 2(2) - \frac{2}{(2)^3} = 4 - \frac{2}{8} = 4 - \frac{1}{4} = \frac{16}{4} - \frac{1}{4} = \frac{15}{4}
$$

So, the slope of the tangent line at the point $$ \left(2, \frac{17}{4}\right) $$ is $$ \frac{15}{4} $$.

### Step 2: Equation of the tangent line

Using the point-slope form of the equation of a line:

$$
y - y_1 = m(x - x_1)
$$

where $$ (x_1, y_1) = \left(2, \frac{17}{4}\right) $$ and $$ m = \frac{15}{4} $$:

$$
y - \frac{17}{4} = \frac{15}{4}(x - 2)
$$

Multiplying through by 4 to eliminate the fraction:

$$
4y - 17 = 15(x - 2)
$$

Expanding this:

$$
4y - 17 = 15x - 30
$$

Rearranging gives:

$$
4y = 15x - 13 \quad \Rightarrow \quad y = \frac{15}{4}x - \frac{13}{4}
$$

### Step 3: Find points A and B

**Point A (x-axis intersection)**: Set $$ y = 0 $$:

$$
0 = \frac{15}{4}x - \frac{13}{4}
$$

Solving for $$ x $$:

$$
\frac{15}{4}x = \frac{13}{4} \quad \Rightarrow \quad 15x = 13 \quad \Rightarrow \quad x = \frac{13}{15}
$$

Thus, point A is $$ \left(\frac{13}{15}, 0\right) $$.

**Point B (y-axis intersection)**: Set $$ x = 0 $$:

$$
y = \frac{15}{4}(0) - \frac{13}{4} = -\frac{13}{4}
$$

Thus, point B is $$ \left(0, -\frac{13}{4}\right) $$.

### Step 4: Calculate the area of triangle OAB

The vertices of triangle OAB are:

- O: $$ (0, 0) $$
- A: $$ \left(\frac{13}{15}, 0\right) $$
- B: $$ \left(0, -\frac{13}{4}\right) $$

The area $$ A $$ of triangle OAB can be calculated using the formula:

$$
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
$$

Substituting the coordinates:

$$
\text{Area} = \frac{1}{2} \left| 0\left(0 + \frac{13}{4}\right) + \frac{13}{15}\left(-\frac{13}{4} - 0\right) + 0\left(0 - 0\right) \right|
$$

This simplifies to:

$$
= \frac{1}{2} \left| \frac{13}{15} \left(-\frac{13}{4}\right) \right| = \frac{1}{2} \cdot \frac{13 \cdot 13}{15 \cdot 4} = \frac{169}{120}
$$

Thus, the exact area of triangle OAB is:

$$
\boxed{\frac{169}{120}}
$$
The exact area of triangle OAB is $$ \frac{169}{120} $$.

5 In this question you must show detailed reasoning.
Fig. 12 shows part of the graph of .
The tangent to the curve at the point meets the -axis at A and meets the -axis
at B .
Ois the origin.
Find the exact area of the triangle OAB .

Upstudy AI Solution

Answer

Solution

Step 1: Find the derivative

Step 2: Equation of the tangent line

Step 3: Find points A and B

Step 4: Calculate the area of triangle OAB

Beyond the Answer

Step 1: Find the derivative of

Step 2: Evaluate the derivative at

Step 3: Write the equation of the tangent line

Step 4: Find the x-intercept (point A)

Step 5: Find the y-intercept (point B)

Step 6: Area of triangle OAB

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Latest Calculus Questions

5 In this question you must show detailed reasoning. Fig. 12 shows part of the graph of . The tangent to the curve at the point meets the -axis at A and meets the -axis at B . Ois the origin. Find the exact area of the triangle OAB .

Upstudy AI Solution

Answer

Solution

Step 1: Find the derivative

Step 2: Equation of the tangent line

Step 3: Find points A and B

Step 4: Calculate the area of triangle OAB

Beyond the Answer

Step 1: Find the derivative of

Step 2: Evaluate the derivative at

Step 3: Write the equation of the tangent line

Step 4: Find the x-intercept (point A)

Step 5: Find the y-intercept (point B)

Step 6: Area of triangle OAB

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Latest Calculus Questions

5 In this question you must show detailed reasoning.
Fig. 12 shows part of the graph of .
The tangent to the curve at the point meets the -axis at A and meets the -axis
at B .
Ois the origin.
Find the exact area of the triangle OAB .